$\lim _{n \rightarrow \infty} \left[ \frac{n}{n^{2}+1^{2}} + \frac{n}{n^{2}+2^{2}} + \ldots + \frac{n}{n^{2}+n^{2}} \right]$ ની કિંમત શોધો.

$\mathop {\lim }\limits_{n \to \infty } {\left\{ {\left( {1 + \frac{{{1^2}}}{{{n^2}}}} \right)\left( {1 + \frac{{{2^2}}}{{{n^2}}}} \right)\left( {1 + \frac{{{3^2}}}{{{n^2}}}} \right) \dots \left( {1 + \frac{{{{(n - 1)}^2}}}{{{n^2}}}} \right)} \right\}^{1/n}}$ ની કિંમત શોધો:

$\lim _{n}$ ${\rightarrow \infty} \frac{1}{n}\left(\frac{1}{e^{1 / n}}+\frac{1}{e^{2 / n}}+\frac{1}{e^{3 / n}}+\ldots+\frac{1}{e^{2n/n}}\right)=$

$\lim _{n}$ ${\rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots \left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}=$

લક્ષની કિંમત શોધો: $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n^2 + r^2}$

જો $\lim _{n \rightarrow \infty} \frac{1}{n} \log \left(\frac{(2 n)!}{n^n \cdot n!}\right)=\int_1^2 f(x) d x$ હોય, તો $f(x)=$